The general class of solutions to the lossless, one-dimensional,
second-order wave equation can be expressed as
where
right-going traveling waves
left-going traveling waves
and where and are assumed twice-differentiable.
This traveling-wave solution of the wave equation was
first published by d'Alembert in 1747.
Notice the traveling-wave solution of the 1-D wave equation has
replaced a function of two variables , by two functions of a
single variable in time units2, greatly reducing computational
complexity.